Question

The equation of motion of a particle starting at $t=0$ is given by $x=5sin(20t+\frac{\pi }{3})$, where $x$ is in centimeter and $t$ is in second. When does the particle come to rest for the second time?

• A. $\frac{\pi }{10}s$
• B. $\frac{7\pi }{100}s$
• C. $\frac{7\pi }{120}s$
• D. $\frac{5\pi }{7}s$

Answer 1

The correct option is C $\frac{7\pi }{120}s$

Given $x=5sin(20t+\frac{\pi }{3})$
By differentiating, velocity is
$\Rightarrow v=100cos(20t+\frac{\pi }{3})$
At $t=0,x=\frac{5\sqrt{3}}{2}cm$ and $v=+50cm{s}^{-1}$.

Thus, it is travelling towards the right extreme position. Velocity $v=0$ happens for the second time when it reaches the left extreme position $x=-5$.
$\Rightarrow -5=5sin(20t+\frac{\pi }{3})$
$\Rightarrow 20t+\frac{\pi }{3}=\frac{3\pi }{2}\Rightarrow 20t=\frac{3\pi }{2}-\frac{\pi }{3}=\frac{(9-2)\pi }{6}$
$\Rightarrow t=\frac{7\pi }{120}.$